Quantum Distribution Functions for Bosons, Fermions, & otherwise Objects

1. Quantum Distribution FunctionsforBosons, Fermions, & otherwise Objects

2. How to do the Lagrange Multiplier technique • function f(x1, x2, …) • Constraint g(x1, x2, …)=c • Form new function F = f + l (g-c) • Maximize/Minimize it wrt x1, x2, … • Choose something for l based upon other information Maximize a function subject to certain constraints on the dependent variables http://www.slimy.com/~steuard/teaching/tutorials/Lagrange.html http://www.cs.berkeley.edu/~klein/papers/lagrange-multipliers.pdf

3. Example Minimize f = xy subject to constraint x2+y2=1 F = xy +l (x2+y2-1) (y+2lx) + (x+2ly) = 0 dF/dx = y + 2lx (1+2l)y + (1+2l)x = 0 dF/dy = x + 2ly y = -x x2+y2=1  x2+x2=1  2x2 = 1  x=±0.707

4. OUTLINE • Harris 8.1 • Quick Basic Probability Ideas • Harris 8.4-8.5 • Derivation of Boltzmann Distribution Fn • How to Use it • How to Normalize it • Harris 8.2-8.3 Macroscopic Descriptions • Entropy and Temperature • Density of States • Harris 8.6 • Comparison of Ytot* Ytot for Bosons & Fermions • Detailed Balance • w/o special requirements • bosons • fermions • Distribution Fns • Interpreatations of the ea factors • Harris 8.7-8.10 • 8.7 Density of States, Conduction Electrons, Bose-Einstein Condensation • 8.8 Photon Gasses • 8. Laser Systems • 8.10 Specific Heats

5. Harris 8.1 Thermodynamic Systems

6. Microinfo vs Macroinfo { ri, pi, Li, Si, Ei,…} i = 1,…, hugen number Etot, P, V, T, N,…

7. Microstates vs Macrostate Very few microstates are similar to this Most Probable arrangement Many microstates are similar to this # collection of most probable microstates should be described with macro-parameters P, V, N, T, Etot,… S = kB ln W

8. Systems where microscopic approach is limited • Solids • 3-body problems in ‘planetary’ motion • Hartree-Fock Procedure • Multielectron atomic structure • Nuclear Structure • e Each system has it’s own macroscopic parameters, but the set always includes Ntot , Etot, Ave Kinetic Energy (~T).

9. What is most likely arrangement of 4 balls in 2 bins?

10. BACKGROUND PROBABILITY IDEAS

11. Given N=5 objects and p=1 bin; How many ways can one put n=2 objects in the bin ? (in a definite order)

12. Given N=5 objects and p=1 bin; How many ways can one put n=2 objects in the bin ? (without regard to order) Note that after filling this box, there are (N-n) objects unused.

13. Given N total objects and p total bins; How many ways can one put n1 objects in bin #1 n2 objects in bin #2 n3 objects in bin #3 * * * (without regard to order) * * *

14. * * * n1 n2 n3

15. Probability of finding a particular arrangement:

16. Probability Summary N total objects p total states np * * * n5 n4 n3 n2 n1

17. Harris 8.4BOLTZMANN DISTRIBUTION Probability of finding a particular energy e subject to the constraint that there are N total particles and Etot energy

18. p * * * 2 1

19. p * * * 2 1

21. Sterling’s Formula The second term is largest by at least a parsec

25. note: reset 1 + a a

28. To discover the expression for the normalization constant (assume smooth spectrum) OK, so what is b = ?

29. Temperature is defined in terms of the average kinetic energy

30. Boltzmann Distribution Probability of finding a particular energy e subject to the constraints that there are N total particles and fixed Etot ea = 1/kT

31. How to normalize theBoltzmann DistributionandDensity of States

32. We normalized the Boltzmann distribution assuming all energies could occur Prob = e-a e-E/kT = A e-E/kT

33. Many, many possible states, closely spaced in energy Just did this

34. Finite number of states,but no restriction on filling * * *

35. Density of States conduction band valence band core electrons

36. Density of States Isolated Na Na as solid Density of states # states per Energy interval = D(E)

37. Calculating Averages if were motivated enough to normalize ND before-hand if decided to do it later

38. Many, many closely-spaced states,but with restriction on filling Just did this

39. Finite number of states,but with restriction on filling * * * w5 w4 w3 w2 w1 D(e) is funky notation, so use wi

41. Harris 8.2-8.3Macroscopic DescriptionsEntropy & Flow of Time James Sethna @ Cornell: Chapter 5 http://pages.physics.cornell.edu/sethna/StatMech/EntropyOrderParametersComplexity.pdf http://www.tim-thompson.com/entropy1.html http://en.wikipedia.org/wiki/Entropy_%28statistical_thermodynamics%29 http://en.wikipedia.org/wiki/Entropy http://en.wikipedia.org/wiki/Ludwig_Boltzmann http://en.wikipedia.org/wiki/Satyendra_Nath_Bose http://en.wikipedia.org/wiki/Fermi_distribution

42. Types of Entropy Entropy is what an equation defines it to be. Wiki: Carnot • Thermodynamic Entropy • Statistical Mechanics Entropy • Information Entropy H (8-3) Wiki: boltz H (8-2) http://en.wikipedia.org/wiki/Entropy_%28statistical_thermodynamics%29

43. Shannon quote (1949): The theory was in excellent shape, except that he needed a good name for “missing information”. “Why don’t you call it entropy”, von Neumann suggested. “In the first place, a mathematical development very much like yours already exists in Boltzmann’s statistical mechanics, and in the second place, no one understands entropy very well, so in any discussion you will be in a position of advantage. Wiki: History_of_Entropy

44. Stat Mech ExampleS = kB ln(#microstates) S = kB ln(4) S = kB ln(1) S = kB ln(6) S = 0 S = 1.38 kB S = 1.79 kB

45. Stat Mech Example pi = 0.167 p1 = 0.25 p3 = 0.25 pi = 1 p2 = 0.25 p4 = 0.25 S=- kB 1 ln(1) = 0 S = 1.79 kB S = 1.38 kB

46. Flow of Time Changes between microstates are generally easily reversible. It is not necessarily likely that changes between macrostates can be reversed .

48. Harris 8.6Back to Probability Distributions

49. COMPARING PROBABILITIES Y*Yfor Indistinguishable Boson/Fermion Particles to thosewithout worrying about B/F requirements OR what do the B/F requirements do to probabilities?

50. Distinguishable Particle Probabilities One particle in a state b Ytot = Yb(1) Prob = Ytot* Ytot = Yb(1) *Yb(1) = 1 Two particles in a state b Ytot = Yb(1) Yb(2) Prob = Yb(1)*Yb(1) Yb(2)*Yb(2) = 1 Three particles in a state b Ytot = Yb(1) Yb(2) Yb(3) Prob = Yb(1)*Yb(1) Yb(2)*Yb(2) Yb(3)*Yb(3) = 1 So what ? Nothing special happens here…..